Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,11,12)(7,8,10,9)(14,18)(15,21)(16,24)(26,28)(29,30,34,33)(31,37,39,35)(32,40,42,36)(38,43,41,44);(5,7)(6,9)(8,12)(10,11)(14,18)(16,24)(19,22)(23,27)(29,31,34,39)(30,35,33,37)(32,41,42,38)(36,44,40,43);(5,8,11,9)(6,7,12,10)(13,14,17,18)(15,22,21,19)(16,20,24,25)(23,28,27,26)(29,31)(30,35)(32,38)(33,37)(34,39)(36,43)(40,44)(41,42);(1,2,3,4)(5,9,11,8)(6,7,12,10)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,27)(25,28)(29,32)(30,36)(31,38)(33,40)(34,42)(35,43)(37,44)(39,41);(5,10,11,7)(6,8,12,9)(13,16)(14,20)(15,23)(17,24)(18,25)(19,26)(21,27)(22,28)(29,33,34,30)(31,35,39,37)(32,40,42,36)(38,43,41,44)Polycyclic presentation:
2^9|a^2=f,b^2=fgh,c^2=fh,d^2=gi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5187762);